Island Study

So today's activity went very well.  The noise level was high ... but all due to constant math talk!  The very talk you want in your classrooms!  (I didn't post a picture with students - no faces allowed but here is a picture of our empty space ... waiting for students to arrive on the islands!)

This idea comes from Cheesemonkey's post ... read it here!  Thank you for sharing.

I am drowning in grading.  Giving feedback is a must!  This process is great for simplifying peer to peer feedback as well as teacher feedback.

To break up the monotony of working in desks, I brought in beach towels - 10 of them ... to create "islands" ... oases of math learning!  On each island there is a group whiteboard, 3 markers, 3 calculators, and necessary handouts (could be done without any but I'll explain why they were helpful).

As students entered they set their bags aside, brought only their math notebooks with them, selected an island and had a seat.  I required no more than 3 people to an island.  I had a few islands with just 2 people.

I gave each team an index card ... students put their names on it.

Then I gave instructions ... our topic of the day is solving systems using matrices.  The first half of class was practicing solving 2 x 2 matrices by hand, the second half of the day was practicing solving 3 x 3 matrices by calculator.  (We have 90 minutes ... every other day ... rotating schedule).

I projected 3 systems on the board.  In groups students solved each system using matrices ... clearly outlining the determinant, inverse matrix, and matrix multiplication.  As groups finished a problem I checked it and hole-punched their index card.  When punched, students immediately moved on to the next problem.  Groups had to rotate who did the writing and who coached.

As students finished the 3 systems that were projected, they moved to a handout with word problems - practicing the same skill but first finding the equations.  So having the handout served as a differentiation tool ... students who worked quickly through the projected problems could move forward.

Students clamored for their hole punches!  Amazing. Free.  Simple!

About half way through we switched gears so that students could learn how to solve systems of equations using their calculators.  To get a hole punch every member of the team had to have the answer on his/her calculator.

Teams earned about 12 - 16 hole punches in the class period.  BUT much more than that ... every student was coached in how to solve the problems of the day.  Every student got feedback.  Every student had support for the parts they found difficult.

And the energy was high.  The conversations totally about math.  And they clamored for their hole punches!  Amazing. Free. Simple!

#MTBoSChallenge: Week 5 Summary

My summary for this week ... 

1 Great Discussion:  

Odd One Out using our parent functions ... activity here and here.  We had fun with this conversation ... and students will be ready for their compare/contrast questions on the test tomorrow!  Students found these ideas:

1. The quadratic function because it is the only one symmetrical about the y-axis.
2. The exponential function because it is the only one with an asymptote.
3. The exponential function because it is the only one that does not pass through the origin.
4. The linear function because it is the only one with an unrestricted range.
5. The linear function because is the only "straight" line.
6. The linear function because it is the only one that can be found in the third quadrant.
7. The square root function because it is the only one that is only found in the first quadrant.
8. The square root function because it is the only one with a restrict domain.

2 Good activities this week: 

We did stations to review for the test coming tomorrow. 

• I grouped the desks in groups of six ... to have five groups in the room.
• I used https://www.superteachertools.net/ to create the goups.
• I used acrylic frames ... one task in each frame.
• Students had one answer sheet they carried from group to group.
• We reviewed transformations, regression equations from data, function notation, domain and range as well as vocabulary.
• Groups were instructed to help one another ... the room buzzed with the happy talk of students working together!

One set of classes had a little extra time due to the way our blocked classes fell.  I used the extra time to explore Desmos' Central Park with students.  Students were thoroughly engaged in the activity!  And it provided a good preview to our next unit in translating problem situations into equations!

3 Things I'm looking forward to this week!

• Students take their first test this week.  I'm eager to see how they do!  The first test has the potential to cause some anxiety!
• We start systems of equations with a heavy emphasis on problem solving.  I'm using the Tortoise, Hare and Rat as the opener!  I also plan to use Mathalicious Flicks.
• Students are going to do their first homework assignment on Desmos this week!  I look forward to teaching them how to do that! (They've seen me model work on Desmos!)

Odd One Out - parent function warm-up

We are about to wrap up our introductory unit on functions.  Our big ideas in the unit are understanding function notation, being able to write domain and range in set and interval notation, recognizing parent functions and their attributes, and transforming parent functions (moving monsters as well).

I'm working on stations for review.  One station will be the following "odd one out" exercise:

The instructions say, "Three of the following functions – A, B, C, and D – have something in common. The other one differs from them in some way. Which one does not belong? Give a reason. Several answers may be possible, but for different reasons. Can you find more than one possible choice?"

I look forward to hearing students' reasonings ... and I wonder if they will have chosen a function with a reason that I have not yet identified!

#70Days Number strings

Math Talk
Number Talk
Strategy Sharing
Number Strings
Cluster Problems
Problem Strings

This morning I checked out the "youcubed" site to see what had been uploaded recently.  On the teachers' page, the first video/article is about number talks.  Since I plan to start the year with these, I've been reading as much as possible on the topic.  I took part in Jo Boaler's course, How to Learn Math last summer and was overwhelmed with making connections to advanced algebra 2 (a new/different curriculum for me).  Now that I have experienced the curriculum and the "typical" student, I am ready to infuse the course with more numeracy activities for sure!

What caught my eye on this site was in the "comments" section of this first article.  Someone shared a site new to me on "Number Strings."  I read Building Powerful Numeracy for Middle and High School Students this summer so I was somewhat familiar with the idea of a number string.  This new site on strings is an excellent resource.  Number strings start with a helper problem - a basic problem in which students share their answer and explain their strategy.  Typically in a string there are additional problems related to the original helper problem.  It might look like this:

3 is 100% of what number?

3 is 50% of what number?

6 is 50% of what number?

3 is 25% of what number?

6 is 25% of what number?

3 is 10% of what number?

6 is 20% of what number?

3.8 is 10% of what number?

This string is longer than some.  It is inspired from a multiple choice test question: 3.84 is 12% of what number?  The choices of answers included:  0.03125, 3.125, 32, or 46.08.  In the discussion online,  a teacher remarked that she didn't want her students to use a calculator or set up an equation.  Instead she wanted to prepare them to answer the test question by working on number strings.

I know I want to work on fraction and decimal skills with my students.  I'm debating about whether to start number talks with whole numbers.  I'm guessing that my 9th and 10th graders have never participated in a number talk ... unless they have participated in our Texas UIL number sense competition training.

Where do I begin?  Will it be OK to jump in with fractions - or will that stymie conversation too much?  Thoughts?

#70Days Sparking mathematical discourse

I have a tendency to frequent the same Half Price Books.  So today we drove northeast to visit a different store.  I love HPB ... each one has it's own personality, it's own special collection of books!

Today I found this book ... and at first I dismissed it totally - I don't need another worksheet book.  But I decided to take a look inside.  I was pleasantly surprised to find activities that require some thought and often have more than one right answer.  I can see using these activities to promote mathematical discourse.

Here is one example. Three of these expressions have something in common. The other one differs from the others in some way. Which one does not belong?  Give a reason!  Several answers may be possible but for different reasons. Can you find more than one possible choice?

Here is another example from that same chapter. In this second example, I am expecting most students to choose "d."  I'll be challenging them to find a different choice! 

There are at least 8 different sets of puzzles in the book. Several will inspire good conversation!  Can't wait to try it out!

Questions and Dialogue

At the beginning of our unit on systems of equations we discussed the question, What is a solution?
We returned to the question as I introduced solving linear inequalities and systems of linear inequalities.  Students participated readily ... less pause than the first time around ...

What is a solution?

Student A: anything that satisfies the equation ...
Me:  could someone add to that thought?
Student B:  it's the numbers that solve the equation ...
Me:  Hmmmm, I don't think we can use the root word in the definition ... any other thoughts?
Student C:  it fits in the equation
Me:  OK ... it satisfies, it solves, it fits ... any other thought?
Student D:  it makes the equation true
Me:  Yes ... to all of the above!
Me:  Now, if I give you an equation with 2 variables, how might you represent the solution visually?
Student E:  could you restate that question?
Me:  (I write on the board) y = 2x + 3; How might I represent the solution to this equation visually?
Student F:  (2, 7)
Me:  OK, that point is true.  How else might I represent the solution visually?
(pause)
Me:  OK ... are there other solutions besides (2, 7)?
Several students:  yes ... many of them ... infinite number
Me:  Yes!  So how might I represent those infinite solutions visually?
Student G:  a line
Me:  Yes!  So how or why does a line represent the infinite solutions to that equation?
Student H:  a line continues in both directions forever
Student I:  a line is made of infinite number of points
Me:  Yes!

We then extended our conversation to include inequalities ... talking about why we shade and what that tells us.

I want to get better at asking questions ... I want to ask better questions.  I also want to develop strong math discourse in our class!

These are the "essential" questions suggested by our district curriculum document:

1. Describe the solution to a system of equations with 3 variables.
2. How do you determine the variables for a system of equations?
3. How do you write a system of equations from information given in a problem situation?
4. How do you know if your solution to a system of equations is reasonable?
5. How do you connect the solution to a system of equations to the context of a problem 
6. situation?
7. How do you decide which method to use to solve a particular system of equations?
8. How do you solve systems with more than two variables?

These questions hit the surface and skim across the topic.  I want something deeper, richer for my students.

Questions and more questions!

Questions and questioning are at the heart of rich math dialogue.  We are in our first week of school and I realize that I need to work on my questioning skills to tap into the strengths of my diverse students.

I have found a few resources to use to help me in preparing worthwhile questions.

1.  At the Ontario Ministry of Education, there is an excellent article entitled, "Asking Effective Questions."  In the article there are eight tips for asking effective questions.  But then there are many, many questions or question stems providing an amazing framework for asking good questions.

2.  I also ran across this book by Schuster and Anderson.  The book is set up by strands in math.  Even though I teach algebra, I find this book useful in developing questions to help my ninth graders make sense of the prerequisite math that they still struggle with.  The questions are open-ended and challenging.  For example:  "Carla and Fiona are having a mathematical debate about the equation y = 1/2 x + 3.  Carla thinks that every time y changes by one, x changes by 2.  Fiona thinks that every time y changes by one-half, x changes by one.  What do you think?"

3.  One other book I ran across provides ideas for differentiating.  It is also organized in strands.  For each topic, the authors, Small and Lin, provide ideas for open questions and 2 parallel questions at different levels.  One example that caught my eye is:  "Another function is a lot like the given function.  What might it be?  Option 1:  y = 3x^2 + 4  Option 2:  y = 3^x.  Follow up questions are provided:  What happens to your function when x gets big?  When x is quite small?  What kinds of things might you change but still have a similar function?  What would you choose not to change?

I'm on the prowl for other great resources for developing strong questioning skills.  Send me a note if you know of others!